A two-inch cube ($2\times2\times2$) of silver weighs 3 pounds and is worth $\$200$. How much is a three-inch cube of silver worth?  Round your answer to the nearest dollar.
Answer: The volume of a two-inch cube is $2^3=8$ cu inches, while that of a three-inch cube is 27 cu inches. Therefore, the weight and value of the larger cube is $\frac{27}{8}$ times that of the smaller. $\$200(\frac{27}{8})=\boxed{\$675}$.